Multivariate polynomial inequalities of different \(L_{p,W}(V)\)-metrics with \(k\)-concave weights
Let \(W\) be a \(k\)-concave weight on an open convex set \(V\) in \(\mathbb {R} ^m,\;k\in [0,\infty]\), and let \(\mu_W\) be the weighted measure on \(V\) generated by \(W\) with \(\mu_W(V)<\infty \). Next, let \(L_{r,W}(V),\;0\leq r\leq \infty \), be the space of all Lebesque-masurable functions \(f:V\rightarrow \mathbb {R}^1\) on \(V\) with the finite functional \(\| f\|_{L_{r,W}(V)}:=(\mu_W (V)^{-1}\int_V| f(x)| ^rW(x)dx)^{1/r}\) for \(0<r<\infty \). For the case \(r=0\) and \(r=\infty \), this functional is naturally extended. By \(\mathcal{P}_{n,m}\) is denoted the set of all algebraic polynomials of \(m\) variables with real coefficients of degree at most \(n\). Lower and upper estimates of a constant \(A_k\) defined by \[ A_k(n,m, p,q):= \sup_V\sup_{W\in C_k(V)}\sup_{P\in \mathcal {P}_{n,m}} \| P\|_{L_{q,W}(V)}/\| P\|_{L_{p,W}(V)},\;0\leq p<q\leq \infty, \] are founded, where \(C_k(V)\) is the set of all \(k\)-concave weights on \(V\) with \(k\in [0,\infty]\). For \(k=0\in (0,\infty ]\), estimates of \(A_k\) are new and they are sharp with respect to \(n\) as \(n\rightarrow \infty \). For \(k=0\), the obtained by A. Brudnyi estimate is improved. Moreover, several examples of computation or estimation for \(\| P\|_{L_{q,W}(V)}/\| P\|_{L_{p,W}(V)}\) are presented.
polynomial inequality of different metrics
scientific article; zbMATH DE number 6806077
scientific article